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Induction in a linear conductor.

§ 28. A closed secondary wire from B will be displaced from $B_{1}$ into position $B_{2}$, while a primary conductor A at the same time passes from position $A_{1}$ to $A_{2}$, and the intensity of the primary current increases from $i_{1}$ to $i_{2}$. At the beginning and the end of time T, in which these processes take place, the two conductors shall be at rest and the primary current shall be constant; if no other electromotive forces acts on B, then this wire will eventually be, as before, without current. We want to determine the quantity of electricity, which has passed in time T through a cross section of the wire, and namely we will only consider the convection current at this place.

After the expiry of the whole process, the surface of B has nowhere a electric charge. It follows that the quantity of electricity that streamed through is the same for all cross sections, and that the conductor can be decomposed into infinitely thin current tubes, so that in each of them and equally through all cross-sections, the same quantity of electricity is streaming.

We consider in detail one of these tubes, and call ds an element of their length, ω is a vertical cross-section, Ndt the number of positive ions which pass through it during the time dt in the assumed positive direction s, N'dt the number of negative ions which move in the opposite direction, e is the charge of a positive and $-e'$ the charge of a negative ion. The total current through ω is then

$i=\int (N\ e+N'\ e')\ d\ t.$

(31)

Furthermore, ${\mathfrak {E}}_{s}$ and ${\mathfrak {E}}'_{s}$ are the electric forces acting in the direction of ds, which come into consideration for a positive or a negative ion. By Ohm's law we shall assume, that the motion of ions by these forces is thus determined, so that N and $N'$ are proportional to its mean value; this and the proportionality to ω, we express by

$N=p{\overline {{\mathfrak {E}}_{s}}}\omega ,\ N'=q{\overline {{\mathfrak {E}}'_{s}}}\omega {,}$

where p and q are constant factors.